has only finitely many rational solutions \(p/q\) , or equivalently the supremum
of the set of numbers \(\mu \)  for which the inequality (REF ) has
infinitely many solutions. If \(\alpha \)  is irrational, then \(\mu \left(\alpha \right) \ge 2\) . If \(\alpha \)  is a real algebraic irrational number, then
\(\mu \left( \alpha \right) =2\)  by Roth's theorem [1]}. If \(\mu \left(\alpha \right) =\infty ,\)  then \(\alpha \)  is called a Liouville number.
